`sin^(2)30^(@)+tan^(2)45^(@)+tan^(2)60^(@)`

Answer

To solve the expression \( \sin^2(30^\circ) + \tan^2(45^\circ) + \tan^2(60^\circ) \), we will follow these steps:

### Step 1: Find the values of the trigonometric functions
- The value of \( \sin(30^\circ) \) is \( \frac{1}{2} \).
- The value of \( \tan(45^\circ) \) is \( 1 \).
- The value of \( \tan(60^\circ) \) is \( \sqrt{3} \).

### Step 2: Calculate the squares of these values
- \( \sin^2(30^\circ) = \left( \frac{1}{2} \right)^2 = \frac{1}{4} \)
- \( \tan^2(45^\circ) = (1)^2 = 1 \)
- \( \tan^2(60^\circ) = (\sqrt{3})^2 = 3 \)

### Step 3: Substitute these values into the expression
Now, we substitute the calculated values into the expression:
\[ \sin^2(30^\circ) + \tan^2(45^\circ) + \tan^2(60^\circ) = \frac{1}{4} + 1 + 3 \]

### Step 4: Simplify the expression
To simplify \( \frac{1}{4} + 1 + 3 \), we first convert \( 1 \) and \( 3 \) into fractions with a common denominator of \( 4 \):
- \( 1 = \frac{4}{4} \)
- \( 3 = \frac{12}{4} \)

Now we can add them:
\[ \frac{1}{4} + \frac{4}{4} + \frac{12}{4} = \frac{1 + 4 + 12}{4} = \frac{17}{4} \]

### Final Answer
Thus, the value of the expression \( \sin^2(30^\circ) + \tan^2(45^\circ) + \tan^2(60^\circ) \) is \( \frac{17}{4} \).

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